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Systèmes de racines sur un anneau commutatif totalement ordonné. (Root systems over a totally ordered commutative ring). (French) Zbl 0721.17019

The author develops a theory of root systems over a totally ordered commutative ring. His theory includes the real root systems associated to the Kac-Moody algebras, the root systems that occur in twisted Chevalley groups and those attached to Coxeter groups. The author defines a prebase of roots over a commutative ring K in which \(2\neq 0\) as a quadruplet \(B=(I,V,{\mathfrak a},{\mathfrak r})\) consisting of an indexing set I, a K-module V, a basis \({\mathfrak a}=(a_ i)_{i\in I}\) of V and a family \({\mathfrak r}\) of automorphisms of V such that for each i in I, \(r_ i\) in \({\mathfrak r}\) is a reflection of the vector \(a_ i\) in \({\mathfrak a}\). This B is called a root basis if in the base A each element of \(\Phi =\{w(a_ i) |\) \(a_ i\) in \({\mathfrak a}\) and w is in the Coxeter group W associated to the Coxeter matrix \((m_{i,j})\) where \(m_{i,j}\) is the order of the product of automorphisms \(r_ i.r_ j\}\) has all its coordinates nonnegative or nonpositive.
The main result is the equivalence of the following assertions: (i) B is a root basis, (ii) for each two element subset J of I, the prebase B restricted to J is a root basis, and (iii) for each w in the Coxeter group W with \(w\neq 1\), one has \(w(C)\cap C=0\) where C denotes the set of linear forms on V which take positive values on all the base vectors \(a_ i\) in \({\mathfrak a}\). The equivalence of (i) and (ii) is the proposition 2.10.
In the last section the author considers a root system B with a group of automorphisms \(\Gamma\) and deduces a new root system \(B^ 0\) such that the Coxeter group \(W(B^ 0)\) is isomorphic to the group of fixed points \(W^{\Gamma}\) of W under the \(\Gamma\)-action. In case \(\Gamma\) is finite, a variant of \(B^ 0\), namely, \(B^ 1\) (see 3.31(b)) gives a root basis which is useful in the construction of twisted Kac-Moody groups.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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